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Đề tài " Quasilinear and Hessian equations of Lane-Emden type "

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: −∆p u = uq + µ, Fk [−u] = uq + µ, u ≥ 0, on Rn , or on a bounded domain Ω ⊂ Rn . Here ∆p is the p-Laplacian defined by ∆p u = div ( u| u|p−2 ), and Fk [u] is the k-Hessian defined as the sum of k × k principal minors of the Hessian matrix D2 u (k = 1, 2, . . . ,. | Annals of Mathematics Quasilinear and Hessian equations of Lane-Emden type By Nguyen Cong Phuc and Igor E. Verbitsky Annals of Mathematics 168 2008 859 914 Quasilinear and Hessian equations of Lane-Emden type By Nguyen Cong Phuc and Igor E. Verbitsky Abstract The existence problem is solved and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type including the following two model problems Apu uq ụ Fk u uq ụ u 0 on R or on a bounded domain Q c Rn. Here Ap is the p-Laplacian defined by Apu div Vu Vu p-2 and Fk u is the k-Hessian defined as the sum of k X k principal minors of the Hessian matrix D2u k 1 2 . n ụ is a nonnegative measurable function or measure on Q. The solvability of these classes of equations in the renormalized entropy or viscosity sense has been an open problem even for good data ụ 2 Ls Q s 1. Such results are deduced from our existence criteria with the sharp PYnonpnts s n q p 11 I nr tbp first. pmin.t.inn And s n q Inr thp sppond exponents s pq tot bile inst equation and s 2kq ioi tne second one. Furthermore a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff s potentials dyadic models and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpelainen and Maly Trudinger and Wang and Labutin. This enables us to treat singular solutions nonlocal operators and distributed singularities and develop the theory simultaneously for quasi-linear equations and equations of Monge-Ampere type. 1. Introduction We study a class of quasilinear and fully nonlinear equations and inequalities with nonlinear source terms which appear in such diverse areas as quasi-regular mappings non-Newtonian fluids reaction-diffusion problems and stochastic control. In particular the following two model equations are of N. P. was supported in part by NSF Grants DMS-0070623 and DMS-0244515. I. V. was supported in part .